Graduate Degree
Programs in
Applied
Mathematics
University of Colorado at Boulder
Academic Year 2005-2006
SUPPLEMENT TO THE
CATALOG
Department of Applied
Mathematics
University of
Colorado
526 UCB
Boulder,
Colorado 80309-0526
Phone: (303) 492-1238
email:
appm_info@colorado.edu
http://amath.colorado.edu/
Introduction
The
Department of Applied Mathematics of the University of Colorado at Boulder
offers a range of courses and research opportunities in several areas
including:
Computational Mathematics
Nonlinear Phenomena
Physical Applied Mathematics
Probability and Statistics
For
more information on these areas and research specialties of the Department and
affiliated faculty, please refer to our web site. The following document provides information on a variety of
topics, ranging from admission procedures to degree requirements. If you have questions that are not
answered by this document, please do not hesitate to contact us.
The
Graduate Committee is responsible for interpretation and administration of the
policies described in this supplement.
By its own initiative or by petition from the student or the student's
advisor, the Graduate Committee may consider exceptions to these policies when
warranted by special circumstances.
The Graduate Committee currently consists of Professors Thomas
Manteuffel (Chair), Jem Corcoran, Keith Julien, Manuel Lladser, and Steve
McCormick.
Admissions
The
admission requirements spelled out below apply equally to the M.S. and Ph.D.
programs. The program in which the
applicant applies for is not,
in and of itself, a factor in decisions regarding admission or support. Applicants should be aware, however,
that both programs are competitive and meeting the requirements does not guarantee
admission. Successful applicants
will, in general, have records considerably stronger in breadth and quality
than these minimum standards suggest.
Both
Domestic and International applications should be sent to:
Graduate Program Assistant
Department of Applied Mathematics
University of Colorado
526 UCB
Boulder, Colorado 80309-0526
General Admission Requirements
Applicants
for graduate study in Applied Mathematics must hold at least a bachelorŐs
degree or its equivalent from an institution comparable to the University of
Colorado in applied mathematics, engineering, mathematics, or a natural
science. Applicants with strong
backgrounds in the physical sciences are encouraged to apply. Applicants should have a significant
background in mathematical coursework and the mathematical maturity to
understand basic concepts in pure and applied mathematics.
Exceptional
students who have some deficiencies in their mathematics background may also be
considered for admission; however, such students will likely have to take some
upper division undergraduate mathematics courses during their first year of
graduate study (see "Provisional Admission" section).
Mathematics Courses
The
applicantŐs academic background should include several semesters of upper
division courses of a mathematical nature beyond calculus, including advanced
calculus and linear algebra.
Additional recommended courses include partial differential equations,
complex analysis, numerical analysis, and perhaps probability and
statistics. These courses need not
be in a mathematics department; however, they should require the mathematical
maturity expected of a strong upper-level mathematics undergraduate.
It
is also recommended that applicants have some computing experience. For example, a working knowledge of a
scientific programming language like FORTRAN, C, C++, or MATLAB is recommended,
and experience with UNIX can be very helpful.
GRE Scores
All
applicants are required to submit scores from the aptitude portion of the
Graduate Record Exam. The subject
portion of the GRE's is not required.
TOEFL Scores
All
foreign applicants are required to submit scores from the Test of English as a
Foreign Language (TOEFL). This
requirement will only be waived when the applicant's native language is English,
or if the applicant has received a degree from an English speaking college in
the United States. Minimum TOEFL
scores are: 173 (computer-based), 500 (paper), or 61 (internet-based –
approximate score). TSE (Test of
Spoken English) is not required.
Provisional Admission
Applicants
having the listed qualifications will, if accepted, be classified as regular
degree students. Applicants with a
few deficiencies in the admission requirements listed above will be considered
on an individual basis and, if accepted, will be given provisional status.
Financial Aid
Financial
aid is available to M.S. and Ph.D. students in the form of teaching
assistantships, research assistantships, and fellowships. Request for financial support should be
included with the application for admission. Applicants are encouraged to apply for need based aid
through the Office of Financial Aid at (303) 492-5091.
Deadlines
Graduate
applications for the next fall semester should be received from domestic
applicants by February 1 of that year and from foreign applicants by December 1
of the previous year. Applicants
are strongly encouraged to apply for fall admission. Due to the structure of the graduate courses, admission to
the graduate programs in spring semesters is discouraged.
ADVISING
Each
new student will be assigned a Faculty Advisor (usually the Chair of the
Graduate Committee) for consultation in planning a sound program of study. Advising includes the courses to be
taken and the areas in which to take the preliminary exams. Incoming students will be prevented
from registering until they have approval from their Faculty Advisor. For Ph.D. students, the Chair of the
studentŐs Thesis Committee will assume the duties of the Faculty Advisor when
it is formed.
M.S.
Degree Requirements
General MS Degree Requirements
The
University of Colorado Catalog gives the general requirements for the degree of
Master of Science in all departments of the University of Colorado. The following is a description of the
requirements that specifically pertain to students pursuing a course of study
leading to the degree of Master of Science in Applied Mathematics. It supplements the requirements in the
Catalog. In all cases not
specifically mentioned below, the general requirements as stated in the Catalog
are understood to apply.
The
M.S. degree can be useful as a stepping stone for any student who is
considering a Ph.D. program here or elsewhere. By the same token, it can serve to increase a Ph.D.
studentŐs options here. However,
the M.S. is unique and an important program in its own right. One of its principal advantages is in
preparation for teaching or industry, which is the genesis of the required
numerical analysis and out-of-department sequences discussed below. It is also a flexible program in that
it supports special interest directions, including the teacher training option
listed below.
Department Course Requirements
The
Department requires a candidate to complete an approved program of study
consisting of at least 30 semester hours.
At least 18 of these 30 hours must be in Applied Mathematics courses at
the 5000 level or above. (Neither 4350/5350,
4360/5360, nor 4720/5720 generally count towards this requirement.) All students must complete two
year-long sequences in applied mathematics. See the "Graduate Courses" section of this
supplement for a list of acceptable 5000-level sequences. Other sequences require Faculty Advisor
approval. If APPM 5600/5610 is not
taken, then the numerical preliminary exam becomes compulsory.
All
students are required to take a year-long sequence in an area where mathematics
has significant applications or from the Math Department (Faculty Advisor
approval required). In addition,
upon approval by petition to the Graduate Committee, up to 6 hours may be taken
in courses at the 4000 level in other departments, provided members of the
graduate faculty teach those courses.
M.S. students must maintain a grade point average of 3.0 or better.
Transfer Credit
MasterŐs
degree students may request a maximum of 9 semester hours to be transferred
from another institution. All
transfer requests must have
approval of both the Graduate Committee and the Graduate School. Work already
applied toward a graduate degree received from CU-Boulder or another
institution cannot be accepted for transfer toward another graduate degree of
the same level at CU-Boulder. For
example, work already applied to meet requirements for a masterŐs degree earned
by a student cannot be used toward a second masterŐs degree from CU. In addition, work completed for a
doctoral degree may not be applied toward a subsequent masterŐs degree.
Foreign Language Requirement
There
is no foreign language requirement for the M.S. program.
Thesis vs. Non-thesis Option
The
MasterŐs degree requirements may be fulfilled by following the requirements for
either the thesis (Plan I) or the non-thesis (Plan II) option (see the Catalog
for details). Students who elect
Plan II must have approval of the Chair of the Graduate Committee.
A
student electing to do a thesis must take 4-6 hours of thesis credit, which
count towards the required 30 hours, and must take an oral comprehensive exam
on his/her thesis work. This will
be administered by a committee consisting of the Faculty Advisor and two other
faculty members, at least one of whom must be a regular or affiliate member of
the Department. The student is
expected to submit a hard copy (bound, on thesis bond) and an electronic copy
(in PostScript or PDF format) of the thesis to the Department before
graduation. The electronic version
will be placed on the Department web site for public access. The department
copy of the dissertation should have title, author, and graduation year
embossed on the spine. To help
cover this additional expense, the department will reimburse the student up to
$30 toward the binding expense.
The Denver Bookbinding Company (http://www.denverbook.com/) provides this service at a reasonable
cost.
A student electing the non-thesis option must pass (Pass or Ph.D. Research Pass) any one of the four Ph.D. preliminary exams. (See the "Ph.D. Degree Requirements" section for more detail).
A
student who fails either the written preliminary exam or the oral thesis
defense may, in a later semester, make one and only one more attempt to satisfy
this requirement. In doing so, the
student may switch between the thesis and the non-thesis option of the program,
or between one preliminary exam area and another.
Adequate Progress
Normally,
a student must be enrolled in at least 5 credits of qualifying courses for each
semester (summer excluded) until the degree is awarded. A student not required to maintain
full-time status may register part-time.
Students who wish may enroll in the Time Out Program through the
Registrar's Office upon approval of the Faculty Advisor. Otherwise, the Graduate Committee may
remove an inactive student from the degree program.
Time Limit
All
requirements for the M.S. degree must be completed within four years of the
start of graduate studies (see the Catalog for details).
Ph.D. Degree Requirements
General Requirements
Studies
leading to the Doctor of Philosophy degree must be chosen so as to contribute
to a high level of scholarship in a broad field of study. Since applied mathematics is by nature
interdisciplinary, these will include courses in one or more application areas
as well as those within the Department. The Catalog of the Graduate School
gives the general requirements for the degree of Doctor of Philosophy. The following information supplements
those requirements. In all cases
not specifically mentioned below, the general requirements as stated in the
Catalog are understood to apply.
The program of study must be approved by the Faculty Advisor.
Program of Study
A
minimum of 60 credits is required for the degree, including 30 in courses
numbered 5000 or above and 30 hours of Applied Math dissertation credit.
(Neither 4350/5350, 4360/5360, nor 4720/5720 generally count towards this
requirement.) A grade of B- or
higher must be attained in each course.
To prepare for the preliminary exams, students are encouraged to take
the graduate sequences in applied analysis (5440/5450) and numerics
(5600/5610). An additional
graduate sequence in an application area is also recommended. See the last section for a list of
acceptable 5000-level sequences.
Other sequences require Faculty Advisor approval.
Students
must take at least two semesters of seminar courses (8000, 8100, 8300 or
8600). These courses are to be
taken no earlier than the second year of graduate study in the Department of
Applied Mathematics. First year
graduate students are not permitted to enroll in these seminar courses except
by special petition to the graduate committee. Note:
Transcripts might state "repeat—not for credit" when
seminar courses are taken more than once.
This statement is an artifact of the system and should be ignored in
this case. Repeated seminars will
be credited towards the M.S. or Ph.D.
Finally,
each student must take a year-long graduate sequence outside of applied
mathematics in an area where mathematics has significant application. Faculty Advisor approval of this
sequence is required.
Dissertation Hours
Students
are required to enroll in a minimum of 30 dissertation hours to complete their
Ph.D. requirements. Students can
enroll in a maximum of 10 dissertation hours per semester, and must be enrolled
in a minimum of 5 dissertation hours after passing the comprehensive exam. (Students electing to be part-time
after passing the comprehensive exam may enroll in 3+ dissertation hours, but
they must be full-time the semester they defend their dissertation.) No more than 10 hours taken before the
comprehensive exam can be counted toward the 30 minimum hours. Students must be enrolled in 5
dissertation hours the semester they defend their dissertation.
Transfer Credit
The
Graduate School will allow doctoral students to transfer from another
institution up to 21 semester hours of course work toward the Ph.D.
degree. All transfer requests MUST
have the approval of the Graduate Committee in Applied Mathematics. Work
already applied toward a graduate degree received from CU-Boulder or another
institution cannot be accepted for transfer toward another graduate degree of
the same level at CU-Boulder. For
example, work already applied to meet requirements for a masterŐs degree earned
by a student cannot be used toward a second masterŐs degree from CU. In addition, work completed for a
doctoral degree may not be applied toward a subsequent masterŐs degree.
M.S. Degree for Ph.D. Students
Courses
taken at the University to satisfy the requirements for the M.S. degree in
Applied Mathematics normally will be counted in considering the minimum
requirements for the Ph.D. degree in Applied Mathematics.
A
student pursuing a Ph.D. degree need not receive the M.S. degree. Any Ph.D. student also wanting to
receive the M.S. degree must satisfy the requirements for that degree. A student may complete the M.S.
non-thesis option by passing one of the preliminary exams. Interested students should contact the
Graduate School for the necessary materials.
Preliminary Exam
Preliminary exams are offered in four areas: analysis, numerics, partial differential equations, and probability/statistics. Students must take the numerics and analysis exams, and either one of the other two. Each is a three-hour written exam. Guidelines and previous exams are available in the departmental office and on the web to aid in preparation.
The
preparatory courses for these prelims are APPM 5440-5450 for analysis, APPM
5600-5610 for numerics, APPM 5470 (not MATH 5470) for partial differential
equations, and APPM 5520-5560 for probability/statistics. Please note that APPM 5520 and APPM
5560 can be taken in any order, and that APPM 5570-5580 is not preparatory for
the probability/statistics prelim.
Students
are expected to complete the preliminary exam requirement successfully prior to
the start of their third year, and prior to passing comps and being admitted to
candidacy. Continuation of
financial support is contingent upon meeting this requirement. Exams are given at the beginning of the
fall and spring semesters.
Foreign Language Requirement
The
Department requires that, before advancing to candidacy, all students must
provide evidence of competency in a foreign language. Approved foreign languages include Chinese, French, German,
Japanese, Russian, and Spanish, but others may be approved by petitioning the
Graduate Committee. Anyone with a
foreign language as a native tongue may be judged by the Graduate Committee to
have met this requirement, while others may satisfy the language requirement by
having successfully completed a one-year college course. Alternatives for establishing
competency may be proposed to the Graduate Committee. In any case, providing evidence of competency is the
student's responsibility.
Thesis Committee
After
choosing a field of specialization, the student will present to the Chair of
the Graduate Committee for approval a list of five faculty members to serve on
her/his Thesis Committee. All
members must have CU graduate faculty status. At least three of the Committee members must be on the
Department faculty; the others can be affiliated faculty members or any other
faculty member outside the Department.
The Graduate Committee and the Dean of the Graduate School must approve
the make-up of the Thesis Committee.
Changes can be made any time to the Committee membership provided the
Graduate Committee and Dean of the Graduate School approve them. The Chair will serve as the student's
thesis advisor and supervise her/his research. The Chair must be a member of the Department faculty or
affiliate faculty, as well as the CU graduate faculty. If an affiliate serves as Chair, then a
member of the Department faculty must serve as an Associate Chair.
Comprehensive Exam (Thesis Proposal)
Prior to admission to candidacy, and not later than the end of the third year in the Department, each student must pass the comprehensive exam. (Students must satisfy the preliminary exam requirement before attempting the comprehensive exam.) The purposes of the comprehensive exam are to insure that the student has a sufficient grasp of the fundamentals of the chosen thesis area to begin research, the ability to exchange ideas and information with the members of the examining board, and a broad base of knowledge in applied mathematics.
At least two weeks before the exam, an application for admission to candidacy for the Ph.D. must be submitted for approval to the Graduate School. At least one week before the exam, the student must submit a 5-10 page thesis proposal, complete with motivation for the topic and references to key papers, to each member of the Thesis Committee. This proposal should be written in consultation with the Chair of the Thesis Committee.
The
exam will consist of a presentation by the student on her/his research proposal
of at most one hour in length, followed by a questioning period of up to one
additional hour. The presentation portion is open to all faculty and students
in the program. The student is
responsible for ensuring general announcement of the comprehensive exam to the
Department at least three days beforehand. The Thesis Committee will constitute the examining
board. A passing grade is given if
at least four of the five members (including the Chair) of the examining
committee vote satisfactory performance.
Thesis Requirement
A
thesis must be based on original investigation and reflect a mature
understanding and critical judgment of the subject matter, as well as
familiarity with tools and methods of research. The thesis subject must be approved by the Thesis Committee,
and must be submitted to the Committee at least 30 days prior to the
defense. The thesis should consist
of material that is publishable in a peer-reviewed journal in, or related to,
applied mathematics, and it must follow the format prescribed by the Graduate
School (see the Catalog).
The
student is expected to submit a hard copy (bound, on thesis bond) and an
electronic copy (in PostScript or PDF format) of the thesis to the Department
before graduation. The electronic
version will be placed on the Department web site for public access. The
department copy of the dissertation should have title, author and graduation
year embossed on the spine. To
help cover this additional expense, the department will reimburse the student
up to $30 toward the binding expense.
The Denver Bookbinding Company (http://www.denverbook.com/) provides this service at a reasonable
cost.
The Thesis Defense
After
the thesis has been completed, a final exam on the thesis and related topics
will be conducted. This exam is
oral and open to the public. The student is responsible for ensuring general
announcement of the thesis defense to the Department at least three days
beforehand. The exam is to be
administered by the Thesis Committee.
More than one dissenting vote will disqualify the candidate in the final
exam.
Time Limit
Doctoral
students are expected to complete all degree requirements within six years of
admission (see the Catalog for details).
Grievance Policy
If a student feels that he/she has received unfair treatment academically or as a teaching or research assistant, then the student should first attempt to resolve the problem with the faculty member involved. If this fails, the student should talk to the Faculty Advisor who will bring the matter before the Graduate Committee. If the student is dissatisfied with this resolution, he/she may file a written formal complaint with the Associate Dean of the Graduate School.
CONCURRENT B.S./M.S. DEGREE
Purpose of the Program
This is a five-year degree program which leads to both a Bachelor
of Science and a Master of Science degree in Applied Mathematics at the end of
the five years. It enables well-qualified and motivated students to experience
graduate-level course-work earlier in their education and to obtain an M.S.
degree in a reduced time period.
Consult the Applied Math Undergraduate Curriculum Guide for more information about the B.S./M.S.
Combined M.S. and M.A. Program with MCD Biology
Purpose of the Program
This Program
offers two Masters degrees, an M.S. in Applied Mathematics and an M.A. in MCD
Biology. This is a three-year interdisciplinary Program. The Department of
Applied Mathematics will be the primary program.
The goal of the Program is to produce well-trained applied mathematics students who are capable of making serious contributions leading to advancements in molecular biology. Such students will be well educated in computational sciences, statistics, probability, and molecular biology.
Admission to the Program
Students are
expected to meet all requirements for admission to the graduate program in the
Department of Applied Mathematics and possess a basic science background
suitable for pursuit of this dual degree. Students are also expected to meet
minimum requirements for admission to the graduate program in MCD Biology. Adequate undergraduate preparation
consists of successful completion of basic courses on cell and molecular
biology. Any student deemed deficient in either area will be required to take
Cell Biology (MCDB 3120) and Molecular Biology (MCDB 3500) after enrollment.
Students will be
required to apply to both programs, with APPM the primary one. Subject to joint recommendation and
approval by APPM and MCDB, incoming students will be admitted to this Dual
Degree Program as a regular part of the Applied Mathematics recruitment
process.
Interested
students should apply to the Applied Mathematics Program and indicate their
interest in the concurrent degree on their application.
The student will be expected to start the programs simultaneously, except in unusual cases when clearly nothing would be compromised. Continuation of the student beyond the first year is subject to approval by APPM and MCDB.
Curriculum
The
required curriculum in Applied Mathematics includes the following 7
three-credit courses: Methods of Applied Mathematics: Partial Differential and
Integral Equations (APPM 5470), Numerical Analysis I and II (APPM 5600 and
5610), Introduction to Mathematical Statistics (APPM 5520), Statistical
Applications Software and Methods (APPM 5580), Numerical Methods for Unconstrained Optimization (CSCI
6676), and two semesters of Independent Study in
Applied Mathematics (APPM 6900). One semester of APPM 6900 (2 credits) for this
Program will focus on a basic study of the principles of genetics. The second semester of APPM 6900 (1
credit) will focus on oral student presentations on thesis research, including
fielding questions, responding to critiques, and presenting background
information. Both sections of APPM 6900 will be arranged in consultation with
the student's Faculty Advisor, who will nominally serve as the course
instructor.
This package of
21 credits provides the necessary background in general applied mathematics,
computational mathematics, and statistics/probability for students to address
challenging problems at the interface of applied mathematics and biology. This
preparation is appropriate for either an academic or commercial setting,
especially in the emerging area of bioinformatics.
In MCD Biology, the core curriculum in this Program consists of 21 credits as follows. A student takes three 3-credit courses, usually in the second year: Cell Structure and Function (MCDB 5210), Gene Expression (MCDB 5230), and Topics in Cell and Developmental Biology (MCDB 5250). In the third year, a student takes either Molecular Genetics (MCDB 5220), or Cell Signaling and Developmental Regulation (CHEM/MCDB 5426). In either case, the student also takes one 3-credit graduate elective in MCDB and 6 credits of Master's Thesis (MCDB 6950; see below). The graduate elective course can be the other of the two required courses mentioned above.
Master's Prelim and Thesis Requirements
Twenty-one
credits of core courses are required within each Department. The proposed MCDB
courses would fulfill the current APPM requirement of an outside sequence and
election of a third course. Similarly, the APPM courses would serve as outside
and elective courses to fulfill MCDB requirements.
The MasterŐs
degree requirements for the APPM M.S. will be fulfilled by the non-thesis (Plan
II) option. The requirements for a thesis (Plan I) program will apply to the
MCDB masterŐs degree. Thesis hours
count only toward MCDBŐs degree. A student must pass the MCDB Prelim (made up
of the exams in MCDB 5210/5230). The student must also successfully complete 6
credits of MCDB 6950 by writing a Master's thesis on original research in an
area at the interface between Applied Mathematics and Molecular, Cellular, and
Developmental Biology.
The Graduate Chairs of both departments must approve successful completion of their respective degree requirements before either degree is confirmed.
Leaving the Program
A student will
be terminated from the dual degree program if they are terminated from either
individual program. The student may petition the APPM Graduate Committee to
remain in the APPM program if he/she was terminated from the MCDB program.
Computational Science and Engineering Track
Purpose of the Program
The purpose of this program is to meet the needs of students who want to learn the basic concepts and skills of Computational Science and Engineering, and then to continue towards a Ph.D. in a discipline outside Applied Mathematics. A student who completes this program successfully will obtain a MasterŐs degree in Applied Mathematics, in the Computational Science and Engineering Track.
The program is
designed to provide interested students with a foundation in computational
mathematics and, at the same time, to allow sufficient latitude for the student
to become proficient in an outside discipline. Approximately half of the
credits for the MasterŐs degree will be taken from a department other than
Applied Mathematics.
Admission to the Program
A student in the
Computational Science and Engineering Track will be enrolled simultaneously in
two graduate programs, one in Applied Mathematics and one in the department
from which the student wishes to receive a Ph.D. An interested student can apply for admission to this Track
either when applying for graduate study at CU, or at any time in the studentŐs
first two years of graduate study.
First-year and second-year graduate students in any of the participating
departments may apply for admission to this program.
Curriculum
The proposed curriculum is flexible in that a student may choose from a set of courses those most useful to the discipline in which the Ph.D. is sought. Different departments will have different sets of appropriate courses. The specific set of courses for each participating department will be specified in a Memorandum of Understanding between that department and Applied Mathematics. Four examples of such course listings are given below, for: (i) Physics; (ii) Astrophysical and Planetary Sciences (APS); (iii) Aerospace Engineering; and (iv) Atmospheric and Oceanic Sciences (PAOS).
Proposed
Curriculum
I. APPM courses - 15 credits:
1. APPM 5600 (3 credits) - Numerical Analysis I
2.
APPM 5610 (3 credits) - Numerical Analysis II
3.
APPM 5440 (3 credits) - Applied
Analysis I
(Students
who have not had the equivalent of MATH 4310 may need to take this course
before taking APPM 5440.)
4.
One of the following choices, to be approved by the outside advisor:
APPM
5470 (3 credits) - Partial Differential Equations
APPM
5460 (3 credits) - Dynamical Systems, Differential Equations, and Chaos
APPM
5560 (3 credits) - Markov Processes, Queues and Monte Carlo Simulations
Other
approved APPM courses
5.
Thesis Option:
APPM
6950 (3 credits) - MS thesis
6.
Non-thesis option
Graduate
electives (6 credits), at least 3 credits in APPM.
(A student taking this option must pass
one of the four APPM preliminary exams.)
II.
CSCI courses - 3 credits:
CSCI
5676 (3 credits) - High Performance Scientific Computing I
(or
other approved CSCI course)
III. Credits in second department - 12 credits approved by outside advisor
Thesis
option: must include 3 credits of MS thesis in outside department
IV.
Students working as TA's in APPM must also take:
APPM 7400 (1 credit) - Teaching and Learning
Seminar
V.
At least 30 graduate credits are required by the Graduate School.
VI. To waive any of the course requirements:
a)
A student
can waive the requirement for APPM 5600 and 5610 by passing the APPM
prelim in Numerical Analysis.
b)
A student
can waive the requirement for APPM 5440 by passing the APPM prelim in
Applied Analysis, or by passing the final
exam in APPM 5440 given that year.
c)
A student
can waive the requirement for APPM 5470 by passing the APPM prelim in
PDEs.
VII. Other recommended courses
APPM
5430 (3 credits) - Applications of Complex Variables
APPM 5480 (3 credits) - Methods of Applied
Mathematics: Approximation Methods
APPM 6610 (3 credits) - Introduction to
Numerical Partial Differential Equations
APPM
7400 (3 credits) - Stochastic Simulations
CSCI 5686 (3 credits) - High Performance
Scientific Computing II
SOME SAMPLE PROGRAMS
A. Computational Physics, non-thesis option
Any four of the following six courses,
along with the required courses in APPM and CSCI:
PHYS 5250-3. Quantum Mechanics I
PHYS 5260-3. Quantum Mechanics II
PHYS
7310-3. Electromagnetic
Theory I
PHYS
7320-3.
Electromagnetic Theory II
PHYS 5210-3. Classical Mechanics
PHYS 7230-3. Statistical Mechanics
B.
Computational
Astrophysics, non-thesis
option
Any four of the following eight courses,
along with the required courses in APPM and CSCI:
ASTR 5110-3. Internal Processes in Gases 1
ASTR 5120-3. Internal Processes in Gases 2
ASTR 5140-3. Astrophysical and Space Plasmas
ASTR 5150-3. Introductory Plasma Physics
ASTR 5400-3. Introduction to Fluid Dynamics
ASTR 5540-3. Mathematical Methods
ASTR 5560-3. Radiative Processes in Planetary Atmospheres
ASTR 5820-3. Origin and Evolution of the Planetary Systems
C.
Computational
Aerospace Mechanics, non-thesis
option
Fall semester,
first year
ASEN 5012-3. Mechanics of Aerospace Structures
APPM 5470-3. Partial Differential Equations
APPM
5600-3. Numerical Analysis I
Spring semester, first year
ASEN 5022-3. Dynamics of Aerospace Structures
APPM 5610-3. Numerical Analysis II
CSCI 5676-3 High Performance Scientific Computing I
Fall semester, second year
ASEN 5007-3. Introduction to Linear Finite Elements
APPM 5440-3. Applied Analysis I
APPM Elective-3.
Spring semester, second year
ASEN Core-3.
ASEN Elective-3
Elective
(any department)-3.
Note: A student may replace APPM 5470 (PDEs) with either APPM 5460
(Dynamical Systems) or APPM 5560 (Markov Processes)
D.
Atmospheric and Oceanic Sciences, non-thesis
option
Any four of the
following eight courses, along with the required courses in APPM and CSCI:
ATOC
5060-3. Dynamics of the Atmosphere
ATOC
5061-3. Dynamics of Oceans
ATOC
5220-3. Nonlinear Dynamics
ATOC
5225-3. Thermodynamics of
Atmospheres and Oceans
ATOC
5400-3. Introduction to Fluid
Dynamics
ATOC
5410-3. Fluid Instabilities,
Waves, and Turbulence
ATOC
5560-3. Radiative Processes in
Planetary Atmospheres
ATOC 6100-3. Predicting Weather and Climate
Master's Requirements
In accordance
with the rules of CUŐs Graduate School, credits from a given graduate course can
be applied only towards one MasterŐs degree. Specifically, if graduate credits from a participating
department are used to obtain a MasterŐs degree in Applied Mathematics in the
Computational Science and Engineering Track, then those same credits cannot
also be used to obtain a MasterŐs degree from the other department. The credits can be applied towards a
Ph. D. from either department.
The program is
self-contained. If desired, a
student can finish with a MasterŐs degree in Applied Mathematics, and embark on
a career at the MasterŐs level.
APPM Teacher Licensure Option
Every
graduate student in the Department of Applied Mathematics takes a year-long
sequence of courses in some area of application of mathematics. One option is to take this sequence in
the School of Education, and ultimately to obtain both a MasterŐs degree in
Applied Mathematics and pursue a license to teach mathematics in a secondary
school (i.e., in middle through high school). This option is not simple, and pursuing it will delay
graduation from the department.
Nevertheless, for graduate students in Applied Mathematics who also seek
a teaching license, here are some guidelines.
The
Teacher Education Program (TEP) in the School of Education for Secondary
Mathematics Teacher Licensure consists of seven courses (EDUC 3013-4, EDUC
3023-4, EDUC 4112-3, EDUC 4122-3, EDUC 4232-3, EDUC 6804-3, EDUC 5375-4), plus one semester of student teaching (which
includes EDUC 4512-2 and 4712-12, and is a full-time full-semester in-school
commitment), plus a passing
score on the PRAXIS II or PLACE licensure exam in mathematics. However, only students who have been
admitted to TEP are eligible to enroll in all but the first of these
courses. Before being admitted to
TEP, a student must have met requirements that include a minimum of 56 hours of
college work, grade point average minimums (2.75 in several areas, including
mathematics courses), 25 hours of youth experience, and a passing score on the
PRAXIS II or PLACE licensure exam in mathematics. Prior to student teaching, the student must have completed
all but the student teaching semester courses, a series of mathematics
requirements, and the PRAXIS II or PLACE Basic Skill Assessment. How much preliminary work will be
required depends on details of the student's previous college background. Interested students should consult with
an advisor in the School of Education in order to learn precise details. A student interested in pursuing this
dual-objective program should plan to complete all of the necessary courses
within APPM in the first two years.
Some of the courses required by the School of Education may also be
taken during this time. With very careful planning from the beginning of
the APPM program, it is potentially possible to complete both the APPM and TEP
within three to three and one-half years.
A student following this plan will not necessarily receive financial assistance (in the
form of a TA) from APPM after the second year.
Interested
students should consult with an advisor in the School of Education in order to
learn precise details. Dana Runge (Education, room 151, 303-492-2559,
Dana.Runge@Colorado.EDU) is the most appropriate contact in the School of
Education. The APPM requirement of
a year-long sequence outside the Department can be satisfied by taking EDUC
4372, plus either EDUC 4232 or EDUC 4122.
The other requirements for a Master of Science in Applied Mathematics
are found in the previous sections of this supplement.
Graduate Courses
Numerous
courses in other departments at the University, as well as the University of
Colorado at Denver, are, in essence, courses in applied mathematics and can be
taken for graduate credit in Applied Mathematics. In fact, each graduate student must take a year-long
sequence outside the department.
Consult the Faculty Advisor for more information and for approval.
Acceptable
5000-level APPM sequences include the following (others require Faculty Advisor
approval):
5430-5470, 5440-5450, 5460-5470, 5470-5480, 5520-5540, 5520-5560, 5570-5580, and 5600-5610.
The following courses that are crosslisted as graduate/undergraduate courses do not count towards the 30 credit hour M.S. or Ph.D. requirement:
APPM 5350 (3) Methods in Applied
Mathematics: Fourier Series and Boundary Value Problems
APPM 5360 (3) Methods In Applied
Mathematics: Complex Variables and Applications
APPM
5720 (3) Open Topics In Applied Mathematics
All of the
remaining courses listed below do
count towards the 30 credit hour MS or Ph.D. requirement.
APPM
5120 (3). Introduction to
Operations Research. Studies linear and nonlinear programming, the
simplex method, duality, sensitivity, transportation and network flow problems,
some constrained and unconstrained optimization theory, and the Kuhn-Tucker
conditions as time permits.
Prereqs.: APPM 3310 or MATH 3130. Same as APPM 4120 and MATH
4120/5120. (Normally offered
spring semester.)
APPM
5380 (3). Modeling in Applied
Mathematics. An exposition of a variety of
mathematical models arising in the physical and biological sciences. Students carry out modeling projects
and present these in class. Topics
can vary: GPS navigation, medical
imaging, ocean waves, computerized facial recognition. Prereqs.: APPM 2350 and 2360. Recommended: APPM 3310, 4350, and
4650. Same as APPM 4380. (Normally offered fall semester.)
APPM 5430 (3). Methods in Applied Mathematics: Applications of Complex Variables. Reviews basic ideas of complex analysis, including solutions of ODE's and PDE's of physical interest via complex analysis; conformal mapping including Schwarz-Christoffel transformations and generalizations; computational methods; Riemann-Hilbert problems; and topics in asymptotic methods. Prereq.: APPM 4360 or 5360, or instructor consent. (Offered on a variable schedule.)
APPM
5440 (3). Applied Analysis 1.
Discusses the elements of basic real and complex analysis, Banach
spaces, LP spaces, and many relevant inequalities. Includes applications of existence and uniqueness of
solutions to various types of ordinary differential equations, partial
differential equations and integral equations. Prereqs.: MATH 4310 and 4320 or equivalent; MATH 3130 or
equivalent; or instructor consent.
(Normally offered fall semester.)
APPM
5450 (3). Applied Analysis 2.
Continuation of APPM 5440. Prereqs.: APPM 5440 or instructor
consent. (Normally offered spring
semester.)
APPM
5460 (3). Methods in Applied
Mathematics: Dynamical Systems and Differential Equations and Chaos. Introduces the theory and applications of dynamical systems
through solutions of differential equations. Covers existence and uniqueness
theory, local stability properties, qualitative analysis, global phase
portraits, perturbation theory, and bifurcation theory. Special topics may
include Melnikov methods, averaging methods, bifurcations to chaos, and
Hamiltonian systems. Prereqs.:
undergraduate courses equivalent to APPM 2360, 3310, and MATH 4310. (Normally offered spring semester of
even years.)
APPM
5470 (3). Methods of Applied
Mathematics: Partial Differential and Integral Equations.
Studies properties and solutions of partial differential equations. Covers methods of characteristics for
partial differential equations, well-posed problems, main results for the
well-known equations, GreenŐs functions, and related integral equations. Prereqs.: APPM 4350 and 4360 or MATH
4430 or equivalent. Same as MATH
5470. (Normally offered fall
semester.)